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Structural Equation Modelling  

Wayne Crawford and Esther Lamarre Jean

Structural equation modelling (SEM) is a family of models where multivariate techniques are used to examine simultaneously complex relationships among variables. The goal of SEM is to evaluate the extent to which proposed relationships reflect the actual pattern of relationships present in the data. SEM users employ specialized software to develop a model, which then generates a model-implied covariance matrix. The model-implied covariance matrix is based on the user-defined theoretical model and represents the user’s beliefs about relationships among the variables. Guided by the user’s predefined constraints, SEM software employs a combination of factor analysis and regression to generate a set of parameters (often through maximum likelihood [ML] estimation) to create the model-implied covariance matrix, which represents the relationships between variables included in the model. Structural equation modelling capitalizes on the benefits of both factor analysis and path analytic techniques to address complex research questions. Structural equation modelling consists of six basic steps: model specification; identification; estimation; evaluation of model fit; model modification; and reporting of results. Conducting SEM analyses requires certain data considerations as data-related problems are often the reason for software failures. These considerations include sample size, data screening for multivariate normality, examining outliers and multicollinearity, and assessing missing data. Furthermore, three notable issues SEM users might encounter include common method variance, subjectivity and transparency, and alternative model testing. First, analyzing common method variance includes recognition of three types of variance: common variance (variance shared with the factor); specific variance (reliable variance not explained by common factors); and error variance (unreliable and inexplicable variation in the variable). Second, SEM still lacks clear guidelines for the modelling process which threatens replicability. Decisions are often subjective and based on the researcher’s preferences and knowledge of what is most appropriate for achieving the best overall model. Finally, reporting alternatives to the hypothesized model is another issue that SEM users should consider when analyzing structural equation models. When testing a hypothesized model, SEM users should consider alternative (nested) models derived from constraining or eliminating one or more paths in the hypothesized model. Alternative models offer several benefits; however, they should be driven and supported by existing theory. It is important for the researcher to clearly report and provide findings on the alternative model(s) tested. Common model-specific issues are often experienced by users of SEM. Heywood cases, nonidentification, and nonpositive definite matrices are among the most common issues. Heywood cases arise when negative variances or squared multiple correlations greater than 1.0 are found in the results. The researcher could resolve this by considering a small plausible value that could be used to constrain the residual. Non-positive definite matrices result from linear dependencies and/or correlations greater than 1.0. To address this, researchers can attempt to ensure all indicator variables are independent, inspect output manually for negative residual variances, evaluate if sample size is appropriate, or re-specify the proposed model. When used properly, structural equation modelling is a powerful tool that allows for the simultaneous testing of complex models.

Article

Survey Design  

Don H. Kluemper

The use of surveys is prevalent in academic research in general, and particularly in business and management. As an example, self-report surveys alone are the most common data source in the social sciences. Survey design, however, involves a wide range of methodological decisions, each with its own strengths, limitations, and trade-offs. There are a broad set of issues associated with survey design, ranging from a breadth of strategic concerns to nuanced approaches associated with methodological and design alternatives. Further, decision points associated with survey design involve a series of trade-offs, as the strengths of a particular approach might come with inherent weaknesses. Surveys are couched within a broader scientific research process. First and foremost, the problem being studied should have sufficient impact, should be driven by a strong theoretical rationale, should employ rigorous research methods and design appropriate to test the theory, and should use appropriate analyses and employ best practices such that there is confidence in the scientific rigor of any given study and thus confidence in the results. Best practice requires balancing a range of methodological concerns and trade-offs that relate to the development of robust survey designs, including making causal inferences; internal, external, and ecological validity; common method variance; choice of data sources; multilevel issues; measure selection, modification, and development; appropriate use of control variables; conducting power analysis; and methods of administration. There are salient concerns regarding the administration of surveys, including increasing response rates as well as minimizing responses that are careless and/or reflect social desirability. Finally, decision points arise after surveys are administered, including missing data, organization of research materials, questionable research practices, and statistical considerations. A comprehensive understanding of this array of interrelated survey design issues associated with theory, study design, implementation, and analysis enhances scientific rigor.

Article

Testing and Interpreting Interaction Effects  

Jeremy F. Dawson

Researchers often want to test whether the association between two or more variables depends on the value of a different variable. To do this, they usually test interactions, often in the form of moderated multiple regression (MMR) or its extensions. If there is an interaction effect, it means the relationship being tested does differ as the other variable (moderator) changes. While methods for determining whether an interaction exists are well established, less consensus exists about how to understand, or probe, these interactions. Many of the common methods (e.g., simple slope testing, regions of significance, use of Gardner et al.’s typology) have some reliance on post hoc significance testing, which is unhelpful much of the time, and also potentially misleading, sometimes resulting in contradictory findings. A recommended procedure for probing interaction effects involves a systematic description of the nature and size of interaction effects, considering the main effects (estimated after centering variables) as well as the size and direction of the interaction effect itself. Interaction effects can also be more usefully plotted by including both a greater range of moderator values and showing confidence bands. Although two-way linear interactions are the most common in the literature, three-way interactions and nonlinear interactions are also often found. Again, methods for testing these interactions are well known, but procedures for understanding these more complex effects have received less attention—in part because of the greater complexity of what such interpretation involves. For three-way linear interactions, the slope difference test has become a standard form of interpretation and linking the findings with theory; however, this is also prone to some of the shortcomings described for post hoc probing of two-way effects. Descriptions of three-way interactions can be improved by using some of the same principles used for two-way interactions, as well as by the appropriate use of the slope difference test. For nonlinear effects, the complexity is greater still, and a different approach is needed to explain these effects more helpfully, focusing on describing the changing shape of the effects across values of the moderator(s). Some of these principles can also be carried forward into more complex models, such as multilevel modeling, structural equation modeling, and models that involve both mediation and moderation.